Del's Monday Poser!

I'd say they all bend.

..or break into bits ! I've seen pics of one of Del's heartbreak over-stretch moments. Left me in therapy for weeks, but for all the wrong reasosn ;p

It's always a pleasure to start a Monday with an infusion of your humour. Beats this sh!t-for-coffee I'm drinking.

oops!

I was going to ask 'what happens if I draw two 1.5" ( :0 metrodecic confusion) bows at the same time' Following that logic, what hapens if I tug on 1000 3/1000" bows at the same time. Not much apart from me filling a wheelie bin with busted polycarb.

It's a trick, innit ? Two 1 1/2 inchers is the same stuff to be bent as one 3"er. You must have specified polycarbonate instead of wood for a reason, and you also want people to backtrack to 2" full length. 2" sounds a bit stiffer than a tapered thing, but it's not strung fairly for comparison.

I'm babbling and my smiley/emoticons aren't working, so Im going for cofee.

They're all drawn to 20". As they all taper to a point, I'd say they all bend the same with the same curve. The wider ones might be harder to bend, but I think the curvature should be the same. Having ZIP experience with bows (I prefer things that go 'BOOM!' myself) I wouldn't know for sure, but that's my gut feel. All the forces I think are in the plane of the curve and slim to none across the width.

"You string it with a 3" brace hight"

I have no idea what that means.

Have you watched Robin Hood ? Any version, even Costner will do. He's humping along ther hills (het, that's a legit word, especiually if you've seen Kev's version). Well, he's got his bow, and it's already taught. Distance between string/catgut and midpoint of bow (when at rest) is his 3 inches.When he gets excited over a deer or something, he pulls it bag to a fairly impressive curvature.

Your other post follows the same logic as mine - copy cat ! I still haven't got my smiley things, but I still have the bang !

My initial logic was to think of Del's 3 incher. The therapist is teaching me how to deal with such thoughts. I can pull it so much, all well and good. If I split it into two 1 1/2 inchers, how can it make a difference. That is possible if you imagine the bow having a curved (longitudinal profile) as if it had been formed around a sphere) . It's not really dimmensional analysis - the later case gives me four pointy bits rather than two.

Del's 2 incher adds a lot more. We no longer know how the force is applied at this end bits.

I cut some stiff card into various widths. Within described terms, it matters not if I bend 1 1/2"or 3 ".

In order that I I don't break anything, It would be more interesting if Del had gone for doubling size. It would at least cut out competing experimenters.

I was writing my post whilst you were posting yours. Look mate, you've five hours on the U.S. - I'm just now getting to bed whilst you're all bright-eyed and...well...bushy-tailed.

...and is that any excuse for no GA ?

Flippin' cat will give an explantiion that leaves us all puzzled.

Amway, it's impossible to know unless one knows how far the bow is shot,

It's fairly irrelevant, but it's the distance between the centre of the string and the centre of the bow when it is strung.

E.G the string is slightly shorter than the bow, so the bow has some curve on it before you start to pull the string.

Del

All this technical jargon. I mean, you could be selling swampland in Florida for all I know.

Don't know. I am more prone to using sniper tactics and weaponry.

When you realize you just got shot but it takes 3 more seconds for the sound to reach you be sure it wasn't one of Dels bows that did it!

I'm tellin' ya. You can't even stand behind Del and be safe. When it lets go, that stuff just flies all over, though I suppose if you stood right behind 'im you could use 'im as a meat-shield.

I've seen pictures of him on the site. I personally would recommend using a small town phone book or the Sunday paper for a shield first.

I remember these problems from school. Isn't one traveling west at 60 MPH while the other is traveling east at 30 MPH?

My hunch (kyphosis?) is that both the tapered ones would be like the side (flatter curved part) of an ellipse (with the wider one being more pronounced), while the one that's the same width all the way along would be closer to circular.

For a given amount of bending moment, the bow will bend more where it is thinnest. So in each tapered case the bow will be bent more at the ends than the bow with constant width. My guess is that there will be no difference between the two tapered version.

Having said that, I don't understand the archery terms any more than I understood the guy who taught me to sail who said "Cut loose the sheet" just before we went for an unexpected swim.

well, first I'd say a parabolic...... but since it has a varied width then I say a hyperbolic.

yes, its a guess.

Here is an interesting paper on the subject of bow design:

http://archery.berkeley.edu/wp-content/uploads/2011/08/Fundamentals-of-the-Design-of-Olympic-Recurve-Bows.pdf

Woman of Nottingham Forest....?

The beauty and brain is sharper and precise than the arrow. Hit me head first baby!

My initial WAG is also that the curves of each of the three bows (their tillers) [I love new terminology] will be identical with identical lengths of draw. That is if the cord (string? or whatever the correct term should be in archery) can handle the tension without excessive stretching or breakage.

However, if this was true then I really doubt that it would be at all considered a poser here. If this was true I would also expect that the bows made over the centuries that mankind has made them would be nothing more than simple notched, bent rods. I know that a bow is certainly not a simple notched, bent rod.

Now that I've debunked my initial guess, I'll start with the bow closest to a simple rod, the uniform 2" bow. I guess that this to tiller will be a catenary curve.

The tapered bows will be more rigid in the middle and will thus not flex as much in the middle. The compression will happen more at the ends of these bows. I have no idea what curve this tiller will be but I'm certain that it is mathematically known. I suspect the rate of taper will have a significant difference in the curve obtained. I suspect the drawing force at this draw length of both tapered bows can be made to be higher than the uniform bow depending on the taper length and taper curve technique.

I'm way outside my mechanical engineering comfort zone already and I do not have access to a CAD design program so I will stop here.

Are we ready for the pics yet?

I'll post 'em tomorrow morning UK...

dunno what that is in dollars

Del

I've spent like 20 hours modelling it and I cannot get it to bend properly, what am I doing wrong.......oh....wait...... the bow string goes in front of the bow, not the side. Never mind.

Good Answer
Just in slightly simpler language:-
1 Circular because the bending force is proportional to the distance from the tip and the strength is proportional to the distance from the tip. (OK a little bit straighter in the middle because the cord is increasingly shorter than the length of plastic)
2. Same as 1.
3. More bent in the middle than the ends. The bending force is greater in the middle but the strength is the same all along. I'm guessing a parabola.

I love a good, simple, straightforward analysis.

GA

Excel does polynomial fits I think. It would be interesting to see how numerically close it is to y = x².

I dare make a small remark since words have a meaning :it is not the "bending force" but the "bending moment".Force is a vector and moment is the product between the force vector and the distance to the considered section.

Maybe

Pics tomorrow

Del

I'm waiting ....

but seriously I would expect a constant cross-section to have a more uniform bend along its length, an dwith a tapered cross section, the larger the cross section in the middle the stiffer it should be and move the curvature further out along the arms .... gut instinct.... I don't have the time or inclination to go looking for formulas .... too busy!

Maybe you're right, but it seems to me that this: "Ration Bending moment to Inertia of section" must have at least one typo in it somewhere.

∞φß

I am guessing at some of the jargon but;

3" and 1.5" tapering to a point will shape the same when drawn the same distance.

2" along the full length will bend only in the idle section so the shape will be different.

Only a gut feeling have no technology to play with on site.

Will keep watching for the answer.

The 2 tapered bows bend exactly the same (within the experimental errors)

The parallel limbed bow bend much more in the middle.

Pics in order, cut out tapered bows, narrow bow, wide bow, parallel bow.

So there you have it.

One thing that makes analysis of a bow tricky is the way the angle of applied force is constantly changing as the bow is drawn.

Gold stars to all who got it right , and those who at least thought about it!

Del

I made some simplified computations:

- compute first the deformed beam with the normal force (parallel to axis)

- compute the further deformation due to the force normal to the axis and

sum the 2

Here are the results for the 2 beam forms:

Bow with constant width

Bow tapered.

The red curves are the deformations under the axial force and the blue are the sum of both deformations.

All computations were made in dimensionless fields since ONLY the form was important.

I dare say they are not very far from what you obtained on the wall, unfortunately living in down town I have not a wall for such empirical analysis.

I would appreciate your comment.

Any way they confirm my first comment (your Pics as well)!

The blue line looks pretty realistic. I think you make the blue line a bit thinner at the end it will flex a bit more as it looks a bit stiff at the end.

I love the whole juxtaposition between theoretical and practical, a 'simple' bow is surprisingly complex... and don't get me started on the boomerang.

I genuinely did the experiment expecting the thinner one to look a bit more like the parallel one. Once I knew the result I could rationalise it. Some things just aren't intuitive.

Of course practicality takes over and in a real bow it has to be fat enough at the tip to attach the string. Although the tips can be surprisingly slim even on heavy bows. It's a common mistake to have the tips too big which robs the bow of speed.

You have to watch the string angle with the tip, that changes all sorts of stuff.
I deliberately chose the draw length to be half the string length.

On a modern target bow they have a long rigid handle (riser) section which gives a string much longer relative to the limbs which changes the angle... mind the limbs are neither straight nor mounted in a straight line.

Simple is good.

I'd love to see an attempt to analyse one of those asymmetric Japanese Yumi bows

Del

I made my simulation for an angle of 57.3°, on your pictures it is (computed with pixels on the picture ) 60°.

I shall compute the forces for the 3 but I am not able to do it before weekend since I live tomorrow early for 2 days. If you measured them it will be interesting to compare theoretical estimations and true measurements.

I'll harvest some dimensions for you. I haven't got a decent protractor, but string length. Nock to nock bow length and draw length should sort you out. I'll annotate them onto a pic.... tomorrow...

Del

Here you can see it's the arc of a circle (within experimental limits) centred on where the string is being pulled.

Circle done in MS Paint.

Del

Simplifying, I'm not going to try the deeper math to find the exact answers.

In both of the shaped bows, the bending moment is proportional to the distance from the end, but so is the section modulus, the bows have the same shape, almost circular.

The 2" strip has the same section modulus full length, stresses will be highest near the middle and lowest at the ends and so will curvature. The bow will do most of its bending in the middle and least at the ends.

Thank you for repeating # 25.

there is only a problem : the bending is NOT linearly proportional to the length along

the beam as the J is.

In fact there is an "almost" proportionality since the bending is proportional to the chord. This is the reason I wrote it is "near to circular".

To find the exact answer the maths are VERY complex since the system of differential equations is NOT linear neither with constant coefficients (as Dell mentions the angle is variable so the ratio of the two forces). It can be integrated but not very easy.

The main problem is that the bending of the force component normal to axis depends on the deflection due to the force component parallel to axis and the projection of the distance due to this second deflection affects the moment of the first force!

I did years ago such an analysis (up to a point since I did not have the time to bring it to its end) but unfortunately I lost all files after a HDD "problem" before I could save it.

The problem can be solved by iterations with much more simple tools using the basic beam equation:

y"= M/(E*J) introducing the functions for M and J and integrating it.

Although this equation is only valid for small deflections it can be used for a good estimation even out of its range.

So, what prompted you in in doing this?

Just to see what happens when you have consistent material?

oh, wait..... I forgot, you're a cat.

I'd been wondering about it from when I was a kid and bought some crossbow plans which had a template for cutting a prod from high tensile aluminium alloy rectangular section bar. How had the guy arrived at the taper? Did it matter?

Wooden bow limbs generally taper in thickness and width, but the curiosity was still there. A question was asked on an archery forum and there was the usual mix of opinion guesswork and being referred to learned papers that obscured a simple question with maths.

So about 45 years later having the scrap material to hand and a bandsaw now... I did the test

Del

Well, You can cross that off your list ........ ^{45 years,.... must be a long list.}

Thanks for the interesting poser.

There is one type of data that I'd like to know from these three "bow" shapes. What was the draw force required for each shape?

The draw on 'em is prob too light to measure decently, but I can have a go tomorrow. For now, suffice it to say the wide taper is heavier draw than the slim one.

Del

Just chipping in in the middle here. From a draw force prespective and speed: What difference does the taper do and is there any? The form of the bow when bend is one thing. Does it take the same force for the same form? Can you achieve the same form at all? And is the speed it retracts back to original form the same?

Questions question questions!

The conclusion is that as long as the limb tapers to zero, the width at the limb root doesn't matter in terms of shape.
The circular (arc of a circle) shape indicates that all parts of the limbs are equally stressed which is a good thing and gives good efficiency.

Extra width does give extra draw weight which equates to arrow speed.

The parallel limb has similar draw weight but has excess weight at the ends of the limbs which will rob them of speed and also has excess stress on the inner limbs which will cause premature failure or mean the same length bow can't be drawn as far.

For a simple bow this arc of a circle is pretty optimal, but in reality, many wooden bows aren't of constant thickness (e.g the English Longbow) another added complication is deflex and reflex which can increase the efficiency of a bow.
I recently built a 35 pound (at 28" draw) bamboo backed yew target bow with a little deflex near the grip and reflex at the outer limb. It was significantly faster than a very similar straight bow of 40 pounds draw (at 28").

Anyone who wants to know more will find plenty by googling Bowyers Diary

Del

Anybody who... you tart ! I refuse to give a link, or even hint to anybody that you run a blog on the topic.

I said half of this before - imagine a globe with longitudes drawn on it. Your nocks are at the poles. No diff between bending 1 x 30 degrees or bending 2 x 15 degrees. The analogy breaks down with that other bow. Blame tyat on Mercator.

Three paragraphs, and I've resisted inuendo !

I don't bother pimping the blog too much these days, it's levelled off at about 8000 hits per month, but a good day gets over 400.

Blimey I've been doing it 4 years now!

Del

rofl

So which is the most popular day on Bowyer's Diary ? Weekend you are probably all off shooting stuff, so I guess posting time is during the week. I seem to have misplaced the link - do you have one ?

Also good info for anyone interested. I'm sure Del knows all this stuff!

http://en.wikipedia.org/wiki/Bow_shape

Some facts and figures, not very accurate as this was only ever a quick experiment.

String length 39"
Bow length (nock to nock) 40.5"

Bracing height 3.5" (This may conflict slightly with the above info, but provides some sort of cross check)

Draw weight at ~20"

Wide bow 0.48 lb

Narrow bow 0.24 lb As one would expect, but I'm surprised it came out as exactly half.

Parallel bow 0.46 lb

Del

And the speed? I get it that the same form will not be achieved!